Preprint:
Hausdorff Dimension of the Contours
of Symmetric Additive Lévy Processes
Davar Khoshnevisan, NarnRueih Shieh, and Yimin Xiao
Abstract.
Let \(X_1\,,\ldots,X_N\) denote N independent,
symmetric Lévy processes on \({\bf R}^d.\)The
corresponding additive Lévy process is
defined as the following
\(N\)parameter random field on \({\bf R}^d\):
$$
X({\bf t}) := X_1(t_1)+\cdots+X_N(t_N)\hskip1in
({\bf t}\in{\bf R}^N_+).
$$
Khoshnevisan and Xiao (2002) have found a necessary and sufficient
condition for the zeroset \(X^{1}\{0\}\) of
\(X\) to be nontrivial with positive probability.
They also provide bounds for the Hausdorff dimension
of \(X^{1}\{0\}\) which hold with positive probability
in the case that \(^{1}\{0\}\) can be non void.
Here, we prove that the Hausdorff dimension of \(X^{1}\{0\}\) is
a constant almost surely on the event \(\{X^{1}\{0\}\cap F\neq\emptyset\}\).
Moreover, we derive a formula for the said constant. This portion of our work
extends the oneparameter formulas of Horowitz (1968)
and Hawkes (1974).
More generally, we prove that for every nonrandom
Borel set \(F\) in \((0\,,\infty)^N\),
the Hausdorff dimension of
\(X^{1}\{0\}\cap F\) is a constant almost surely
on the event \(\{X^{1}\{0\}\cap F\neq\emptyset\}\).
This constant is computed explicitly in many cases.
Keywords.
Additive Lévy processes, level sets, Hausdorff dimension
AMS Classification (2000)
Primary. 60G70
Secondary. 60F15
Support. Research supported in part by a grant from
the National Science Foundation grant DMS0404729.
Pre/EPrints. This paper is available in
Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 841120090, U.S.A.
davar@math.utah.edu

NarnRueih Shieh
Department of Mathematics,
National Taiwan University
Taipei 10617, Taiwan
shiehnr@math.ntu.edu.tw

Yimin Xiao
Department of Statistics and Probability,
A413 Wells Hall
Michigan State University
East Lansing, MI 48824, USA
xiao@stt.msu.edu 
Last Update: September 25, 2008
© 2006  Davar Khoshnevisan, NarnRueih Shieh, and Yimin Xiao